\contentsline {section}{\tocsection {}{1}{Complex abelian integral on elliptic curves}}{2}{section.1}
\contentsline {subsection}{\tocsubsection {}{1.1}{Building blocks of functions on ${\mathbb {C}}$ associate to a lattice}}{2}{subsection.1.1}
\contentsline {subsection}{\tocsubsection {}{1.2}{Abel theory}}{3}{subsection.1.2}
\contentsline {subsection}{\tocsubsection {}{1.3}{Rational differential forms on $E$}}{3}{subsection.1.3}
\contentsline {subsection}{\tocsubsection {}{1.4}{Algebraic universal extension}}{5}{subsection.1.4}
\contentsline {subsection}{\tocsubsection {}{1.5}{Weil pairing}}{7}{subsection.1.5}
\contentsline {section}{\tocsection {}{2}{Complex abelian integral on algebraic curves}}{8}{section.2}
\contentsline {subsection}{\tocsubsection {}{2.1}{Algebraic curve over ${\mathbb {C}}$}}{8}{subsection.2.1}
\contentsline {subsection}{\tocsubsection {}{2.2}{Differential forms}}{10}{subsection.2.2}
\contentsline {section}{\tocsection {}{3}{$p$-adic fields}}{11}{section.3}
\contentsline {subsection}{\tocsubsection {}{3.1}{$p$-adic number}}{11}{subsection.3.1}
\contentsline {subsection}{\tocsubsection {}{3.2}{No $2\pi i$ in ${\mathbb {C}}_p$}}{14}{subsection.3.2}
\contentsline {subsection}{\tocsubsection {}{3.3}{$p$-adic logarithm}}{14}{subsection.3.3}
\contentsline {subsection}{\tocsubsection {}{3.4}{Cyclotomic extension}}{15}{subsection.3.4}
\contentsline {section}{\tocsection {}{4}{Fontaine's rings and $p$-adic Galois representations}}{16}{section.4}
\contentsline {subsection}{\tocsubsection {}{4.1}{$p$-rings}}{16}{subsection.4.1}
\contentsline {subsection}{\tocsubsection {}{4.2}{Teichm\"uller representatives}}{16}{subsection.4.2}
\contentsline {subsection}{\tocsubsection {}{4.3}{The ring $\setbox \z@ \hbox {\frozen@everymath \@emptytoks \mathsurround \z@ $\textstyle E$}\mathaccent "055D{E}^+$}}{18}{subsection.4.3}
\contentsline {subsection}{\tocsubsection {}{4.4}{The ring $\setbox \z@ \hbox {\frozen@everymath \@emptytoks \mathsurround \z@ $\textstyle A$}\mathaccent "055D{A}^+=W(\setbox \z@ \hbox {\frozen@everymath \@emptytoks \mathsurround \z@ $\textstyle E$}\mathaccent "055D{E}^+)$}}{19}{subsection.4.4}
\contentsline {subsection}{\tocsubsection {}{4.5}{The ring $B_{{\mathrm {dR}}}^+$ and the field $B_{\mathrm {dR}}$}}{19}{subsection.4.5}
\contentsline {subsection}{\tocsubsection {}{4.6}{$p$-adic Galois representation}}{21}{subsection.4.6}
\contentsline {section}{\tocsection {}{5}{$p$-adic abelian integral}}{22}{section.5}
\contentsline {subsection}{\tocsubsection {}{5.1}{Lubin-Tate formal groups}}{22}{subsection.5.1}
\contentsline {subsection}{\tocsubsection {}{5.2}{Periods of Lubin-Tate groups}}{25}{subsection.5.2}
\contentsline {subsection}{\tocsubsection {}{5.3}{$p$-adic integration}}{27}{subsection.5.3}
\contentsline {subsection}{\tocsubsection {}{5.4}{$p$-adic periods of abelian integrals}}{29}{subsection.5.4}
\contentsline {subsection}{\tocsubsection {}{5.5}{$p$-adic Riemann relations}}{30}{subsection.5.5}
\contentsline {subsection}{\tocsubsection {}{5.6}{One example of application}}{31}{subsection.5.6}
